Research On Pareto Optimality In Cooperative Game

Game theory deals with cooperation or competition between/among players.According to whether the players can reach a binding agreement,the game is divided into the cooperative game and the noncooperative game.Pareto optimality plays a crucial role in analyzing the cooperative game.Over the past few decades,the Pareto optimality has been widely used to analyze various economic models,such as optimal economic growth,environmental economics and so on.In addition,it was also applied to the research of many control theories,for example,motor optimization design,component coordination control,route optimization and so on.It should be noted that most of the existing works are about the Pareto optimality of the deterministic continuous-time systems or the Pareto optimality for the regular convex linear quadratic(LQ)case.Thus,it should be studied for the nonregular convex case and more systems.Utilizing the theory and methods of optimal control,the main contributions of this thesis are summarized as follows:1.Necessary/sufficient conditions for the existence of the Pareto solutions in a finite horizon cooperative stochastic differential game are studied.Utilizing the necessary and sufficient characterization of the Pareto optimality and the stochastic maximum principle,the necessary conditions for the existence of the Pareto solutions are put forward.Under certain convex assumptions,it is shown that the necessary conditions are also sufficient ones.In addition,we discuss the LQ case in terms of the fixed initial state and the arbitrary initial state,respectively.2.The LQ optimal control problem of the stochastic singular systems is studied in finite horizon.By introducing a new kind of generalized differential Riccati equations(GDREs),the sufficient conditions for the well-posedness of the above optimal control problem are put forward.In addition,the LQ Pareto game of the stochastic singular systems is studied in finite horizon.It is shown that under the solvability of the corresponding GDREs,all Pareto efficient strategies can be obtained by solving a weighted sum optimal control problem.3.The LQ optimal control problem of the stochastic singular systems is studied in infinite horizon.By the equivalent transformation method,the necessary and sufficient condition for the well-posedness is put forward.In addition,the LQ Pareto game of the stochastic singular systems is studied in infinite horizon.By the discussion of the convexity of the cost functionals,a sufficient condition under which the Pareto efficient strategies are equivalent to the weighted sum optimal controls is provided.4.Pareto game of the nominal mean-field stochastic systems is studied.By the discussion of the convexity of the cost functionals,it is shown that the Pareto optimality is equivalent to the weighted sum minimization under the existing conditions.On account of the equivalence,the Pareto-based guaranteed cost control(GCC)problem of the uncertain mean-field stochastic systems is studied.This problem is solved by the GCC of the weighted sum objective functional.Applying the KKT conditions,the necessary conditions for the existence of the Pareto-based guaranteed cost controllers are put forward and all controllers are obtained.In addition,an LMI-based approach is introduced to reduce greatly the computational complexity.5.Necessary/sufficient conditions for the existence of the Pareto solutions in a finite horizon cooperative difference game are studied.Utilizing the necessary and sufficient characterization of the Pareto optimum,this problem is transformed into a set of constrained optimal control problems with a special structure.Employing the discrete version of Pontryagin's maximum principle,the necessary conditions for the existence of the Pareto solutions are derived.Under certain convex assumptions,it is shown that the necessary conditions are sufficient too.In addition,the LQ case is discussed in terms of the fixed initial state and the arbitrary initial state,respectively.